6 - Transfinite Iteration

Summary

This chapter takes the iterative construction of initial algebras into the transfinite, generalizing work in Chapters 2 and 4. It begins with a brief presentation of ordinals, cardinals, regular cardinals, and Zermelo’s Theorem: Monotone functions on chain-complete posets have least fixed points obtainable by iteration. When a category has colimits of chains, if an endofunctor preserves colimits of chains of some ordinal length, then the initial-algebra chain converges in the same number of steps. We discuss the precise length of that iterative construction. We introduce the concept of smooth monomorphisms, providing a relation between iteration inside a subobject poset and in the ambient category. We prove the Initial Algebra Theorem: Under natural assumptions related to smoothness, the existence of a pre-fixed point of an endofunctor guarantees the existence of an initial algebra.